Finite Group DatabasebetaFinite Group Databasebeta| order | 576 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C9×H(5,𝔽2) ⋊ C2 |
| Sylow | C9 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×D9 |
| commutator, cosocle | C2×C9 | C25 |
| Frattini, socle | C2×C3 | C24×D3 |
| Fitting | C9×H(5,𝔽2) | C2 |
| hypercenter | H(5,𝔽2) | D9 |
| hypocenter | C9 | #585 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2 ◁ H(5,𝔽2) | |
1215 conjugacy classes of subgroups; only the 511 normal subgroups are shown.
This group has 1215 conjugacy classes of subgroups — too many to draw. Only its 511 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^2|c^2=f,g^3=h,e^a=ef,g^a=g^2h^2,h^a=h^2,d^b=df,e^c=ef102 conjugacy classes — view the full character table.
| maximal subgroup of | #11864938, #11864953, #11865127, … |
| maximal normal subgroup of | #11864938, #11864953, #11865127, … |
| maximal quotient of | #11900108, #11900114, #11900157, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(576, 5023);